Difference between revisions of "2001 AIME II Problems/Problem 5"
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== Problem == | == Problem == | ||
− | A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets <math>\{4, 5, 6, \ldots, n\}</math> of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of <math>n</math>? | + | A [[set]] of positive numbers has the ''triangle property'' if it has three distinct elements that are the lengths of the sides of a [[triangle]] whose area is positive. Consider sets <math>\{4, 5, 6, \ldots, n\}</math> of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of <math>n</math>? |
− | == Solution == | + | == Solution 1 == |
− | Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset <math>\mathcal{S}</math>. Without loss of generality, consider any <math>a, b, c \,\in \mathcal{S}</math> with <math>a < b < c</math>. <math>\,\mathcal{S}</math> does not possess the triangle property, so <math>c \geq a + b</math>. We use this property to build up <math>\mathcal{S}</math> from the smallest possible <math>a</math> and <math>b</math>: | + | Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset <math>\mathcal{S}</math>. Without loss of generality, consider any <math>a, b, c \,\in \mathcal{S}</math> with <math>a < b < c</math>. <math>\,\mathcal{S}</math> does not possess the [[triangle inequality|triangle property]], so <math>c \geq a + b</math>. We use this property to build up <math>\mathcal{S}</math> from the smallest possible <math>a</math> and <math>b</math>: |
+ | <cmath>\mathcal{S} = \{\, 4,\, 5,\, 4+5, \,5+(4+5),\, \ldots\,\} = \{4, 5, 9, 14, 23, 37, 60, 97, 157, 254\}</cmath> | ||
− | + | <math>\mathcal{S}</math> is the "smallest" ten-element subset without the triangle property, and since the set <math>\{4, 5, 6, \ldots, 253\}</math> is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property. Thus, our answer is <math>n = \fbox{253}</math>. | |
− | + | ==Solution 2== | |
− | + | I claim that the answer is <math>253</math>. We will show that no number greater than 253 works. Consider the subset<math> {4, 5, 9, 14, 23, 37, 60, 97, 157, 254}</math>. This subset does not have the property. It is easy to see that any number less than or equal to <math>253</math> works, and hence our answer is <math>253</math> | |
− | <math>\mathcal{S}</math> is the "smallest" ten-element subset without the triangle property, and since the set <math>\{4, 5, 6, \ldots, 253\}</math> is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property. | ||
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− | Thus, <math>\fbox{ | ||
== See also == | == See also == | ||
{{AIME box|year=2001|n=II|num-b=4|num-a=6}} | {{AIME box|year=2001|n=II|num-b=4|num-a=6}} | ||
+ | |||
+ | [[Category:Intermediate Combinatorics Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 13:27, 2 September 2020
Contents
Problem
A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of ?
Solution 1
Out of all ten-element subsets with distinct elements that do not possess the triangle property, we want to find the one with the smallest maximum element. Call this subset . Without loss of generality, consider any with . does not possess the triangle property, so . We use this property to build up from the smallest possible and :
is the "smallest" ten-element subset without the triangle property, and since the set is the largest set of consecutive integers that does not contain this subset, it is also the largest set of consecutive integers in which all ten-element subsets possess the triangle property. Thus, our answer is .
Solution 2
I claim that the answer is . We will show that no number greater than 253 works. Consider the subset. This subset does not have the property. It is easy to see that any number less than or equal to works, and hence our answer is
See also
2001 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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